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Quadratic addition rules for quantum integers
| Content Provider | CiteSeerX |
|---|---|
| Author | Kontorovich, Alex V. Melvyn Nathanson, B. |
| Abstract | Abstract. For every positive integer n, the quantum integer [n]q is the polynomial [n]q = 1 + q + q2 + · · · + qn−1. A quadratic addition rule for quantum integers consists of sequences of polynomials R ′ = {r ′ n (q)} ∞ n=1, S ′ = {s ′ n (q)} ∞ n=1, and T ′ = {t ′ m,n (q)} ∞ m,n=1 such that [m + n]q = r ′ n (q)[m]q + s ′ m (q)[n]q +t ′ m,n (q)[m]q[n]q for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials F = {fn(q)} ∞ n=1 that satisfy the associated functional equation fm+n(q) = r ′ n (q)fm(q) + s ′ m (q)fn(q) + t ′ m,nfm(q)fn(q). 1. Quantum addition rules Let N denote the set of positive integers. Let K[q] denote the ring of polynomials with coefficients in a field K. For every positive integer n, the quantum integer [n]q is the polynomial [n]q = 1 + q + q 2 + · · · + q n−1 ∈ K[q]. |
| File Format | |
| Journal | J. Number Theory |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Quantum Integer Quadratic Addition Rule Positive Integer Considers Sequence Quantum Addition Rule Let Associated Functional Equation Fm Polynomial Fn Complete Classification |
| Content Type | Text |
| Resource Type | Article |
